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arXiv:2110.05725 · Physical Review A

Thermodynamic contacts and breathing-mode physics of one-dimensional p -wave Fermi gases in the high-temperature limit

Jeff Maki

Published Dec 20, 2021• 5 citations
arXivHTMLPDF

Abstract

An important tool for understanding the effects of interactions in harmonically trapped atomic gases is the examination of their collective modes. One such mode is the breathing or monopole mode, which is special as it is constrained to occur at twice the harmonic trapping frequency when the interactions are scale invariant. When the interactions are not scale invariant, the frequency of the breathing mode will deviate from twice the trap frequency. The deviation itself depends on the thermodynamic contacts, which describe how the energy changes with the interactions. In this work I examine how the thermodynamic contacts and the breathing-mode frequency of a spin-polarized one-dimensional (1D) $p$-wave Fermi gas depend on the 1D scattering volume $\ensuremath{\ell}$ and the effective range $r$ in the high-temperature limit. Such dynamics can be studied in experiments and provide a tool for understanding how the dynamics depends on interactions with a finite effective range.

DOI
10.1103/physreva.104.063314

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An important tool for understanding the effects of interactions in harmonically trapped atomic gases is the examination of their collective modes. One such mode is the breathing or monopole mode, which is special as it is constrained to occur at twice the harmonic trapping frequency when the interactions are scale invariant. When the interactions are not scale invariant, the frequency of the breathing mode will deviate from twice the trap frequency. The deviation itself depends on the thermodynamic contacts, which describe how the energy changes with the interactions. In this work I examine how the thermodynamic contacts and the breathing-mode frequency of a spin-polarized one-dimensional (1D) ppp-wave Fermi gas depend on the 1D scattering volume \ensuremathℓ\ensuremath{\ell}\ensuremathℓ and the effective range rrr in the high-temperature limit. Such dynamics can be studied in experiments and provide a tool for understanding how the dynamics depends on interactions with a finite effective range.
Inspired by the renewed experimental activities on ppp-wave resonantly interacting atomic Fermi gases, we investigate theoretically some experimental observables of such systems at zero temperature in two dimensions, using both mean-field theory and Gaussian pair fluctuation theory. These observables include the two ppp-wave contact parameters and the breathing mode frequency, which can be readily measured in current cold-atom setups with 40K^{40}\mathrm{K}40K and 6Li^{6}\mathrm{Li}6Li atoms. We find that the many-body component of the two contact parameters exhibits a pronounced peak slightly above the resonance and consequently leads to a dip in the breathing mode frequency. In the resonance limit, we discuss the dependence of the equation of state and the breathing mode frequency on the dimensionless effective range of the interaction ${k}_{F}{R}_{p}\ensuremath{\ll}1$, where kF{k}_{F}kF​ is the Fermi wave vector and Rp{R}_{p}Rp​ is the effective range. The breathing mode frequency ${\ensuremath{\omega}}_{B}$ deviates from the scale-invariant prediction of ${\ensuremath{\omega}}_{c}=2{\ensuremath{\omega}}_{0}$, where ${\ensuremath{\omega}}_{0}$ is the trapping frequency of the harmonic potential. This frequency shift is caused by the necessary existence of the effective range. In the small range limit, we predict that the mode frequency deviation at the leading order is given by $\ensuremath{\delta}{\ensuremath{\omega}}_{B}\ensuremath{\simeq}\ensuremath{-}({\ensuremath{\omega}}_{0}/4){ln}^{\ensuremath{-}1}({k}_{F}{R}_{p})$.
The contact is an important concept that characterizes the universal properties of a strongly interacting quantum gas. It appears in both thermodynamic (energy, pressure, etc.) and dynamic quantities (radio-frequency and Bragg spectroscopies, etc.) of the system. Very recently, the concept of contact was extended to higher partial waves; in particular, the ppp-wave contacts have been experimentally probed in recent experiments. So far, discussions on ppp-wave contacts have been limited to three dimensions. In this paper, we generalize the ppp-wave contacts to two dimensions and derive a series of universal relations, including the adiabatic relations, high-momentum distribution, virial theorem, and pressure relation. At the high-temperature and low-density limit, we calculate the ppp-wave contacts explicitly using virial expansion. A formula which directly connects the shift of the breathing-mode frequency and the ppp-wave contacts is given in a harmonically trapped system. Finally, we also derive the relationships between interaction parameters in three- and two-dimensional Fermi gases and discuss possible experimental realization of a two-dimensional Fermi gas with ppp-wave interactions.
For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently, the frequency of the breathing mode ${\ensuremath{\omega}}_{B}$ of the 2D Fermi gas can be different from $2{\ensuremath{\omega}}_{0}$, with ${\ensuremath{\omega}}_{0}$ the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for ${\ensuremath{\omega}}_{B}$. We further calculate ${\ensuremath{\omega}}_{B}$ through the Euler equations in the hydrodynamic regime. The obtained value of ${\ensuremath{\omega}}_{B}$ satisfies the upper bounds and shows deviation from $2{\ensuremath{\omega}}_{0}$, which can be as large as about 88%8.
We examine the breathing mode of a strongly interacting two-dimensional Fermi gas and the role of temperature on the anomalous breaking of scale invariance. By calculating the equation of state with different many-body TTT-matrix theories and the virial expansion, we obtain a hydrodynamic equation of the harmonically trapped Fermi gas (with trapping frequency ${\ensuremath{\omega}}_{0}$) through the local density approximation. By solving the hydrodynamic equations, we determine the breathing mode frequencies as a function of interaction strength and temperature. We find that the breathing mode anomaly depends sensitively on both interaction strength and temperature. In particular, in the strongly interacting regime, we predict a significant downshift of the breathing mode frequency, below the scale invariant value of $2{\ensuremath{\omega}}_{0}$, for temperatures of the order of the Fermi temperature.
We present an experimental investigation of collective oscillations in harmonically trapped Fermi gases through the crossover from two to three dimensions. Specifically, we measure the frequency of the radial monopole oscillation or breathing mode in highly oblate gases with tunable interactions. The breathing mode frequency is set by the adiabatic compressibility and probes the thermodynamic equation of state. In 2D, a dynamical scaling symmetry for atoms interacting via a δ potential predicts the breathing mode to occur at exactly twice the harmonic confinement frequency. However, a renormalized quantum treatment introduces a new length scale which breaks this classical scale invariance resulting in a so-called quantum anomaly. Our measurements deep in the 2D regime lie above the scale-invariant prediction for a range of interaction strengths providing evidence for the quantum anomaly and signifying the breakdown of an elementary δ-potential model of atomic interactions. By varying the atom number we can tune the chemical potential and see the breathing mode frequency evolve smoothly between the 2D to 3D thermodynamic limits.
A quantum anomaly manifests itself in the deviation of the breathing mode frequency from the scale invariant value of 2ω in two-dimensional harmonically trapped Fermi gases, where ω is the trapping frequency. Its recent experimental observation with cold atoms reveals an unexpected role played by the effective range of interactions, which requires a quantitative theoretical understanding. Here we provide accurate, benchmark results on a quantum anomaly from a few-body perspective. We consider the breathing mode of a few trapped interacting fermions in two dimensions up to six particles and present the mode frequency as a function of scattering length for a wide range of effective range. We show that the maximum quantum anomaly gradually reduces as the effective range increases while the maximum position shifts towards the weak-coupling limit. We extrapolate our few-body results to the many-body limit and find a good agreement with the experimental measurements. Our results may also be directly applicable to a few-fermion system prepared in microtraps and optical tweezers.